how to find the surface area of a triangular prism

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A prism is a three-dimensional shape with two parallel, congruent bases.^[1] In a triangular prism, the bases are triangles. A triangular prism also has three lateral sides. To find the surface area of triangular prism, you first need to find the area of the lateral sides, then you need to find the area of the bases. Finally, you need to add these two areas together to find the total surface area. These steps are represented by the formula ${\text{surface area}}=L+2B$ , where $L$ equals the lateral area of the prism and $B$ equals the area of one base.

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Calculate the perimeter of one base. The base is a triangle, so it will have three sides. The area of the perimeter of a triangle is ${\text{Perimeter}}=a+b+c$ , where $a$ , $b$ , and $c$ are the length of each side of the triangle.^[4] It doesn't matter which base you use to calculate, because the two bases of a prism are congruent.
- For example, if the base has three sides measuring 6 cm, 5 cm, and 4 cm, to calculate the perimeter, you would add up all three sides: $6+5+4=15$ . So, the perimeter of one base is 15 cm.
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Plug the perimeter into the lateral area formula. Make sure you substitute for the variable $P$ in the formula.
- For example, $L=15h$ .
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Plug the height of the prism into the lateral area formula. The height of the prism is the same as the length of the side of any lateral face that is not connected to the base. Usually (but not always) this will be the longer side of the lateral face.
- For example, if the height of the prism is 9 cm, your formula will look like this: $L=15(9)$ .
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Multiply the perimeter of one base with the height of the prism. The result will give you, in square units, the lateral surface area of the prism. This is the first value you need to find the total surface area of the prism, so set this value aside while you calculate the area of the base.
- For example, $15(9)=135$ , So, the lateral surface area of the prism is 135 square centimeters.
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Plug the base of the triangle into the formula. Don't confuse the base for another side of the triangle. The base is the side perpendicular to the height.
- For example, if the base of the triangle is 6 cm, your formula will look like this: $A={\frac {1}{2}}6h$ .
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Plug the height of the triangle into the formula. Multiply the base by the height. Then, take half of this value. This will give you the area of the base, in square units. This is the second value you need to calculate the total surface area of the prism.

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Set up the formula for finding the surface area of a prism. The formula is $SA=L+2B$ , where $SA$ equals the surface area of the prism, $L$ equals the lateral area of the prism, and $B$ equals the area of one base.^[6]
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Plug the lateral area into the formula. This is the surface area of all sides of the prism that are not the base. You should have calculated this previously. Make sure that you substitute the lateral area for the variable $L$ .
- For example, if the lateral area of your triangular prism is 135 square centimeters, your formula will look like this: $SA=135+2B$ .
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Plug the area of one base into the formula. Make sure you use the area of just one base, not the total area of both bases combined. Substitute the base area for the variable $B$ .
- For example, if the area of one base of your prism is 9.9 square centimeters, your formula will look like this: $SA=135+2(9.9)$ .
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Complete the calculations. Multiply the area of the base by 2, then add the lateral area. This will give you the total surface area, in square units, of your triangular prism.

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Question

Can you make a clearer description of how to find the surface area of a triangular prism? I am so confused about how to do it, and I am freaked out that I will fail the test in school because of it!

A triangular prism is just a box with triangles for the top and bottom. The walls of the box are the same as they are for any other box - rectangular. To find the surface area of the entire prism, you just find the surface area of the top, the bottom, and each wall - then you add them all together. So focus on each step on its own, as if it were its own problem, and write down the answer. What is the surface area of the triangular top? (Write that down.) What is the surface area of the triangular bottom? (Write that down.) Do the same for each wall, then add them all together. Then you're done! You'll find shortcuts along the way, but if they're confusing you now, don't worry about them.
Question

How can I find the surface area of the triangular prism without knowing its height?

It is not possible to find the surface area of a prism without knowing the height of the prism. If you are given the volume of the prism, it might be possible to derive the height from the formula V = bh, where V equals the volume, b equals the area of the base, and h equals the height. If you know the volume and the area of the base, you can solve for h. From here you can go on to calculate the surface area using the steps above.
Question

I'm very confused on how to find the 2B, can you help?

2B refers to the total area of the two triangular bases. As with any triangle, calculate the area of either triangular base by multiplying the "base" of the triangle (the length of one of its sides) by its height (the perpendicular distance from that side to the opposite vertex) and then dividing by 2. That gives you the area of either triangular base (B) of the prism. Multiply by 2 to get 2B, the total area of the two triangular bases. To that area you then add the "lateral" area (L) of the prism to get the total surface area (L + 2B).
Question

Can you explain the difference between a triangular prism vs. right triangular prism?

The only difference is that while a triangular prism has a base consisting of any triangle, a right triangular prism has a base consisting specifically of a right triangle. In either case the volumes are calculated the same way, as are the surface areas.
Question

Is triangular prism the same with right prism?

"Prism" as used here implies a "right prism." Thus, a triangular prism has two bases, each consisting of a triangle, each congruent with the other, and each perpendicular to the planes of the prism's sides.
Question

When I have the radius of a circle, how do I find the total surface area?

There's no such thing in relation to a circle. A circle merely has an "area," calculated by πr², where r is the radius.
Question

How would I find the area of two right angled triangle prisms that are pasted together?

You would have to know how they are pasted together. Are they pasted together along a lateral side, or along a base? You should calculate the surface area of each prism separately, using the steps above. Then, combine their surface area. Finally, subtract the area of the sides or bases that are pasted together.
Question

What is the relationship between the surface area and volume if the dimensions/measurements are doubled?

For any triangular or rectilinear prism, if each dimension is doubled, the surface area increases by a factor of four, and the volume increases by a factor of eight.
Question

If the rectangular plot one of its side is 24 m and the length of diagonal is 26 m, how do I find the area of the plot?

By the Pythagorean theorem, the square of the diagonal minus the square of one side equals the square of the other side. Thus, 26² - 24² = 676 - 576 = 100 = 10², so that the other side measures 10 m, which means the plot's area is 24 x 10 = 240 square meters.
Question

How do I calculate the volume in liters of a triangular prism?

Calculate the volume in whatever units you're given, then use the appropriate conversion factor to convert to liters.

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Article SummaryX

To find the surface area of a triangular prism, use the formula Surface Area = L + 2B, where L is the lateral area and B is the area of the base. Find the lateral area by calculating the perimeter of the base and multiply it by the height of the prism. Then, find the base area by multiplying the base by the height of the triangle and dividing by 2. Next, plug the lateral area and base area into the Surface Area formula. Multiply the area of the base by 2 and add the lateral area to get your answer. Be sure to label your answer with the proper units squared. If you want to learn how to find the height of the base triangles, keep reading the article!

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